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We consider the isentropic Navier-Stokes-Korteweg equations with anon-decreasing pressure on the whole space $\mathbb{R}^n$ $(n \ge 2)$, wherethe system describes the motion of compressible fluids such as liquid-vapormixtures with phase transitions including a variable internal capillarityeffect. We prove the existence of a unique global strong solution to the systemin the $\mathrm{L}^p$-in-time and $\mathrm{L}^q$-in-space framework, especiallyin the maximal regularity class, by assuming $(p, q) \in (1, 2) \times (1,\infty)$ or $(p, q) \in \{2\} \times (1, 2]$. We show that the system isglobally well-posed for small initial data belonging to $\mathrm{H}^{s + 1, q}(\mathbb{R}^n) \times \mathrm{H}^{s, q} (\mathbb{R}^n)^n$ provided $s > n/q$ if$q \le n$ and $s \ge 1$ if $q > n$. Our results allow the case when thederivative of the pressure is zero at a given constant state, that is, thecritical states that the fluid changes a phase from vapor to liquid or fromliquid to vapor. The arguments in this paper do not require any exactexpression or a priori assumption on the pressure.

Global existence of the Navier-Stokes-Korteweg equations with a non-decreasing pressure in $\mathrm{L}^p$-framework